1. Field of the Invention
The present invention relates to a creation method of a table, a creation apparatus, a creation program and a creation program storage medium, and particularly to a creation method of a conversion table in code conversion or a code book table, or a table used for a layout of experimental design or the like, a creation apparatus, a creation program as software for its operation, and a storage medium of the program.
2. Related Art
First, the basic property of a Latin square and a Latin cube. The Latin square is conventionally known (for example, see non-patent document 1, non-patent document 2, non-patent document 3). That is, each element of a set A (a1, . . . , an) consisting of n symbols is used n times, and the n2 elements in total are arranged as a square having n rows and n columns, and when each element of A appears once on each row and each column, it is called a Latin square on A or an n-order Latin square.
When both the first row and the first column are natural permutations, it is called a reduced or standard Latin square. When the number thereof is denoted by L2(n), the total number of the n-order Latin squares becomes n!•(n−1)!•L2(n). When n is 9 or less, that is, n is 1 to 9, L2(n) becomes as follows.    L2(1)=1    L2(2)=1    L2(3)=1    L2(4)=4    L2(5)=56    L2(6)=9,408    L2(7)=16,942,080    L2(8)=535,281,401,856    L2(9)=377,597,570,964,258,816.
Besides, the basic form of a four-order Latin square in the case of A={1, 2, 3, 4} becomes as shown in FIG. 7(b), and since any of array elements on the first row and the first column are ascending sequences (natural permutations) of 1 to 4, it belongs to the standard Latin square shown in FIG. 8. Incidentally, in FIG. 8, it is assumed that a value of an array element of (•) is set to an arbitrary value of 1 to 4 forming the Latin square. A regular creation method of the above two-dimensional Latin square is opened to the public by the present inventor (see patent document 1).
Incidentally, the basic form of a four-order Latin square in the case of A={0, 1, 2, 3} is as shown in FIG. 7(a). Besides, the symbol is not limited to a numeral, but may be an alphabet or another symbol, and when the alphabet of letters a to d are used as the symbols to form the set A={a, b, c, d}, the basic form of the four-order Latin square is as shown in FIG. 7(c).
According to the Latin square creation method disclosed in the patent document 1, a position of a first element of the Latin square to be created is made the first row and first column, the movement direction of a position of an element to be created is previously set to either one of a column direction in which it is moved in a direction indicated by an arrow in FIG. 9(a) and a row direction in which it is moved in a direction indicated by an arrow in FIG. 9(b), and each element is successively created.
Here, for example, in the case where each element is created in the column direction according to the patent document 1 at each position of four rows and four columns of the four-order standard Latin square on A={0, 1, 2, 3}, as shown in FIG. 10(a), after a value “0” of a first element is set at the first row and first column, when an element (symbol) is successively arranged along each row at each position so that it does not become the same symbol as an already determined array element at the former position in the same row and the same column, the element is regularly arranged in sequence shown in the drawings (b) to (p).
Besides, the above patent document 1 also discloses a method of creating a new Latin square from an existing Latin square. That is, a method is such that the order of the Latin square to be created, and permutations of symbols of the order, and a selecting sequence in accordance with the permutations are determined, a return is successively made along a row and a column from the final position of the row and the column of the existing Latin square to a position where a symbol lower in the selecting sequence than a symbol of an existing array element can be selected, and a symbol of an array element is successively selected at each position along the column or the row from that position to the final position so that it does not become the same symbol as an already determined array element at the former position in the same row and the same column.
According to this Latin square creation method, for example, a return is successively made from the final position at the fourth row and fourth column of an existing four-order Latin square shown in FIG. 11(a) to a position where a symbol “1” lower in the selecting sequence than a symbol “0” can be selected as shown in the drawing (b), the next symbol “1” is arranged at that position, and in the following, a symbol of an array element is successively selected at each position to the final position along a row or a column of the drawing so that it does not become the same symbol as an already determined array element at the former position in the same row and the same column, and each element is determined in the sequence shown in the drawings (c), (d), (e), (f) and (g), and finally, a new Latin square shown in the drawing (g) is created.
On the other hand, each element of a set A (m1, . . . , mn) consisting of n symbols is used n times, and the n3 elements in total are arranged as a cube having n elements in each of three directions (X axis (vertical) direction, Y axis (horizontal) direction, Z axis (depth) direction), and when each element of A appears once on each direction, that is, when n n-order Latin squares which do not have the same value at the respective same positions are overlapped with each other, it is called a Latin cube on A or an n-order Latin cube.
A regular creation method of the three-dimensional Latin cube is invented by the present inventor and is opened to the public (see patent document 2). Although the number of standard Latin cubes was not opened to the public before the publication of the patent document 2, according to the patent document 2, when it is expressed as L3(n), the total number of the n-order Latin cubes is expressed as n!•(n−1)!•(n−1)!•L3(n). When n is 5 or less, the value of L3(n) is L3(1)=1, L3(2)=1, L3(3)=1, L3(4)=64, L3(5)=40246, and the total number of the Latin cubes in each order is 1 in first order, 2 in second order, 24 in third order, 55296 in fourth order, and 2781803520 in fifth order.
However, although the creation method and creation apparatus of the Latin square as the two-dimensional table, and the creation method and creation apparatus of the Latin cube as the three-dimensional table have been invented by the present inventor and are opened to the public in the patent document 1 and the patent document 2, a creation method and a creation apparatus of a similar table having four dimensions or higher are not known, its illustration is also difficult, and there is no generalized creation method independent of dimensions, and therefore, there has been a problem that the use value and use effect of a table can not be further raised.
[Non-Patent Document 1]
Edited by Japan Mathematical Society, “Iwanami Sugaku Jiten”, third edition, Iwanami Shoten
[Non-Patent Document 2]
Koichi Yamamoto, “Various Phases of Latin Square”, Surikagaku, Kabusikikaisha Science, 1979, June, Vol. 17, No. 6, p. 62-66
[Non-Patent Document 3]
Koihi Yamamoto, New Mathematics Lecture “Combinatorial Mathematics”, Asakura Shoten
[Patent Document 1]
JP-A-10-105544
[Patent Document 2]
JP-A-2000-285101